Rule-Level Verification of Graph Transformations for Invariants Based on Edges’ Transitive Closure

Abstract

This paper develops methods to reason about graph transformation rules for proving the preservation of structural properties, especially global properties on reachability. We characterize a graph transformation rule with an applicability condition specifying the matching conditions of the rule on a host graph as well as the properties to be preserved during the transformation. Our previous work has demonstrated the possibility to reason about a graph transformation at rulelevel with applicability conditions restricted to Boolean combinations of edge expressions. We now extend the approach to handle the applicability conditions containing transitive closure of edges, which implicitly refer to an unbounded number of nodes. We show how these can be internalized into a finite pattern graph in order to enable verification of global properties on paths instead of local properties on edges only.

Correctness by construction and style preserving reconfigurations of system of systems

“…They are used to generate code or to transform a graphical model into a more formal one, for example. Endogenous transformations [5][6][7] remains the model invariant but change the state of a system. Vertical transformations [2][3][4]7], e.g.…”

“…refinements, modify the granularity and the level of precision with which the system is represented. Horizontal transformations [5,6] are usually related to model changes or system evolutions.…”

“…Furthermore roll-backs may have to be applied if it is discovered that the system is in an inconsistent state. Transformation checking [5,6], consists in verifying in run-time that a transformation do not introduce any inconsistency. While it is generally more effective than the previous solution, roll-backs are not dismissed.…”

“…It may be the preservation of the system behaviour [2,3], for code generation of model modification, for example. In some other case, correctness is linked to the presence, the absence or the the conservation of some property [6]. Architectural styles are particularly relevant when considering dynamic system.…”

Verifying Monadic Second-Order Properties of Graph Programs

“…The formalism is more expressive than MSO logic on graphs (it is able, for example, to express node-counting MSO properties such as "the graph has an even number of nodes" [21]) but it is not yet clear whether an effective construction for weakest liberal preconditions exists. Percebois et al [15] demonstrate how one can verify global invariants involving paths, directly at the level of rules. Rules are modelled with (a fragment of) first-order logic on graphs in the interactive theorem prover Isabelle.…”